which is obtained by writing the amplitude for the four tetrahedra and then applying the Biedenharn-Elliott identity and the orthogonal relations of \(\lbrace 6j\rbrace \) -symbols.
The divergence comes from \(\sum _{J}d_{J}^2=\delta _{\mathrm {SU}(2)}(\mathbb {I})\)  whose degree is related to the topology of \({\mathcal {M}}\)  (see [1]}, [2]}, [3]} for discussion). Regularization was originally performed by introducing a cut-off on spin \(J\)  [4]}. It was then realized that the divergence is correspondent to the \({\mathfrak {su}}(2)\)  gauge that generates the translational symmetry of the internal vertex thus the regularization can be performed by a partial gauge fixing procedure [5]}, [6]}.
